A pre-symplectic form is just a closed 2-form of constant rank. For instance, the restriction of a symplectic form to a coisotropic submanifold (such as the zero level set of a moment map) is pre-symplectic. Given a pre-symplectic from \(\eta \)  of rank \(k\) , we constructed in [1]} an algebraic structure that encodes the deformations of \(\eta \) , i.e., the 2-forms nearby \(\eta \)  (in the \(C^0\) -sense) which are both closed and of constant rank \(k\) . As in many deformation problems, this algebraic structure is an \(L_{\infty }\) -algebra, which we call the Koszul \(L_{\infty }\) -algebra of \(\eta \) . Its construction – which is somewhat involved due to the simultaneous presence of the closedness and constant rank conditions – relies on a certain \(BV_{\infty }\) -algebra structure on the differential forms and builds on the work of Fiorenza–Manetti [2]}. The Koszul \(L_{\infty }\) -algebra has the property that its Maurer–Cartan elements are in bijection with the pre-symplectic deformations of \(\eta \) .
